Risk Management
Importing the Data
# import data
# returns are in %
# sum return percentages by day
# assumed equal investment weight in each firm
read_func <- function(value, I)
{
filename = paste(value, ".csv", sep="")
ret = read.csv(filename, header=TRUE)
firms = ret[["PERMNO"]]
count = length(unique(firms))
port = count*I
ret_sum = aggregate(.~DATE, data=ret, FUN=sum)
ret_sum = ret_sum[order(as.Date(ret_sum$DATE, "%m/%d/%Y"), decreasing=FALSE),]
ret_date = unique(ret_sum[["DATE"]])
keeps = c("RET","DATE")
ret_sum = ret_sum[keeps]
# gets average
i = 1
while(i < nrow(ret_sum))
{
ret_sum[i,"RET"] = ret_sum[i,"RET"]/count
i = i + 1
}
data = list("portfolio" = port, "returns" = ret_sum)
return(data)
}
invest = 1000000 # investment per firm
file1 = "returns_main"
file2 = "returns_comp"
file3 = "returns_main_hist"
file4 = "returns_comp_hist"
# reads files, gets list of returns and value
data_m = read_func(file1,invest)
data_c = read_func(file2,invest)
data_mh = read_func(file3,invest)
data_ch = read_func(file4,invest)
Histogram of Returns
# histogram function with normal dist overlay
vec_histogram <- function(x, names)
{
h = hist(x, breaks=(length(x)/50), col="red", xlab=names$xlabel,
main=names$title)
xfit = seq(min(x),max(x),length=40)
yfit = dnorm(xfit, mean=mean(x), sd=sd(x))
yfit = yfit*diff(h$mids[1:2])*length(x)
lines(xfit, yfit, col="blue", lwd=2)
}
# lists with labels for histograms
hist_label_m = list("title" = "Main Period Returns w/ Normal Curve",
"xlabel" = "Daily Returns")
hist_label_c = list("title" = "Comparison Period Returns w/ Normal Curve",
"xlabel" = "Daily Returns")
# write histograms for historical returns
vec_histogram(data_m$returns[["RET"]], hist_label_m)
vec_histogram(data_c$returns[["RET"]], hist_label_c)
VaR, $VaR and Expected Shortfall
#function to calculate var
var_calc <- function(returns,port,a)
{
r_vec = returns[["RET"]]
vol = sd(r_vec)
cumdist = qnorm(1-a,0,1)
var = abs(quantile(r_vec,1-a))
dol_var = abs(quantile(r_vec,1-a)*port)
exp_short = vol*dnorm(cumdist)/(1-a)
data = list("VaR"=var, "$VaR"=dol_var,
"Expected_Shortfall" = exp_short)
return(data)
}
# confidence interval for var
conf = 0.95
# var calculations
var_m = var_calc(data_m$returns,data_m$portfolio,conf)
var_c = var_calc(data_c$returns,data_c$portfolio,conf)
# prints out var Variables
var_m
## $VaR
## 5%
## 0.0102009
##
## $`$VaR`
## 5%
## 2040179
##
## $Expected_Shortfall
## [1] 0.01635318
var_c
## $VaR
## 5%
## 0.008870062
##
## $`$VaR`
## 5%
## 283842
##
## $Expected_Shortfall
## [1] 0.01550245
Daily Models
# risk metrics one day modeling
oneday_f <- function(returns,hist_returns,a)
{
# initial variance using historical data
# variance based off of previous 10 days
hist = hist_returns[["RET"]]
variance_0 = var(hist[length(hist)-10:length(hist)])
# variables for while loop
lamda = 0.94
i = 1
variance = c(0)
var = c(0)
exp_short= c(0)
cumdist = qnorm(1-a,0,1)
while(i <= nrow(returns))
{
variance_1 = lamda*variance_0 + (1-lamda)*((returns[i,"RET"])^2)
variance[i] = variance_1
var[i] = -1*sqrt(variance_1)*cumdist
exp_short[i] = sqrt(variance_1)*dnorm(cumdist)/(1-a)
variance_0 = variance_1
i = i + 1
}
returns$variance = variance
returns$VaR = var
returns$ExpShort = exp_short
return(returns)
}
oneday_m = oneday_f(data_m$returns,data_mh$returns,conf)
oneday_c = oneday_f(data_c$returns,data_ch$returns,conf)
# garch model
garch_f <- function(returns,hist_returns,a)
{
# initial variance using historical data
# variance based off of previous 10 days
hist = hist_returns[["RET"]]
variance_0 = var(hist[length(hist)-10:length(hist)])
# function to solve for garch model variables
x.g = garchFit(~garch(1,1),returns[["RET"]])
# summary(x.g)
coef(x.g)
# variables for loop and garch model
i = 1
variance = c(0)
var = c(0)
exp_short= c(0)
cumdist = qnorm(1-a,0,1)
alpha = coef(x.g)[3]
beta = coef(x.g)[4]
omega = coef(x.g)[2]
while(i <= nrow(returns))
{
variance_1 = omega + beta*variance_0 + alpha*((returns[i,"RET"])^2)
variance[i] = variance_1
var[i] = -1*sqrt(variance_1)*cumdist
exp_short[i] = sqrt(variance_1)*dnorm(cumdist)/(1-a)
variance_0 = variance_1
i = i + 1
}
returns$VaR = var
returns$ExpShort = exp_short
returns$variance = variance
return(returns)
}
garch_m = garch_f(data_m$returns,data_mh$returns,conf)
##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: norm
## h.start: 2
## llh.start: 1
## Length of Series: 2767
## Recursion Init: mci
## Series Scale: 0.007927996
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.32110904 0.321109 0.0321109 TRUE
## omega 0.00000100 100.000000 0.1000000 TRUE
## alpha1 0.00000001 1.000000 0.1000000 TRUE
## gamma1 -0.99999999 1.000000 0.1000000 FALSE
## beta1 0.00000001 1.000000 0.8000000 TRUE
## delta 0.00000000 2.000000 2.0000000 FALSE
## skew 0.10000000 10.000000 1.0000000 FALSE
## shape 1.00000000 10.000000 4.0000000 FALSE
## Index List of Parameters to be Optimized:
## mu omega alpha1 beta1
## 1 2 3 5
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3748.5188: 0.0321109 0.100000 0.100000 0.800000
## 1: 3735.3607: 0.0321103 0.122741 0.0967439 0.811486
## 2: 3726.3122: 0.0321096 0.121029 0.0739978 0.799681
## 3: 3725.3035: 0.0321096 0.126442 0.0752300 0.805061
## 4: 3724.6123: 0.0321098 0.121272 0.0712490 0.809206
## 5: 3724.4035: 0.0321100 0.115851 0.0732870 0.814325
## 6: 3723.9937: 0.0321098 0.114808 0.0679513 0.819821
## 7: 3723.9352: 0.0321098 0.113600 0.0678421 0.819764
## 8: 3723.8892: 0.0321098 0.113488 0.0682050 0.820917
## 9: 3723.7844: 0.0321098 0.111549 0.0674358 0.822160
## 10: 3720.7091: 0.0321075 0.0789351 0.0513750 0.872125
## 11: 3719.6966: 0.0321056 0.0694367 0.0466286 0.886533
## 12: 3598.5937: 0.0318379 1.00000e-06 0.00453679 0.996889
## 13: 3571.3127: 0.0318379 1.00000e-06 0.00432796 0.996348
## 14: 3565.7068: 0.0318379 1.00000e-06 0.00488800 0.996199
## 15: 3563.4444: 0.0318025 1.00000e-06 0.00482748 0.995847
## 16: 3560.9257: 0.0317659 1.00000e-06 0.00501565 0.995948
## 17: 3559.5364: 0.0316923 1.00000e-06 0.00515500 0.995639
## 18: 3557.8900: 0.0315436 1.00000e-06 0.00533711 0.995742
## 19: 3557.1529: 0.0313948 1.00000e-06 0.00537041 0.995583
## 20: 3556.4992: 0.0310972 1.00000e-06 0.00552383 0.995613
## 21: 3555.9031: 0.0305018 1.00000e-06 0.00556246 0.995464
## 22: 3555.3510: 0.0293109 1.00000e-06 0.00572724 0.995467
## 23: 3554.8663: 0.0269292 1.00000e-06 0.00581758 0.995294
## 24: 3554.6053: 0.0221658 1.00000e-06 0.00606972 0.995246
## 25: 3554.5320: 0.0201772 1.00000e-06 0.00610699 0.995109
## 26: 3554.3335: 0.0221659 1.00000e-06 0.00610391 0.995166
## 27: 3554.1144: 0.0261431 1.00000e-06 0.00613606 0.995076
## 28: 3553.7914: 0.0340975 1.00000e-06 0.00609815 0.995168
## 29: 3553.5406: 0.0340975 1.00000e-06 0.00656361 0.994848
## 30: 3553.5246: 0.0340975 1.00000e-06 0.00656750 0.994868
## 31: 3553.5127: 0.0340975 1.00000e-06 0.00652820 0.994876
## 32: 3553.5037: 0.0341001 1.00000e-06 0.00653122 0.994890
## 33: 3553.4981: 0.0341051 1.00000e-06 0.00650697 0.994894
## 34: 3553.4936: 0.0341154 1.00000e-06 0.00650847 0.994904
## 35: 3553.4902: 0.0341360 1.00000e-06 0.00649107 0.994907
## 36: 3553.4871: 0.0341772 1.00000e-06 0.00649168 0.994915
## 37: 3553.4839: 0.0342596 1.00000e-06 0.00647828 0.994917
## 38: 3553.4800: 0.0344245 1.00000e-06 0.00647788 0.994923
## 39: 3553.4610: 0.0364572 1.00000e-06 0.00646208 0.994930
## 40: 3553.4572: 0.0370594 1.00000e-06 0.00643008 0.994953
## 41: 3553.4572: 0.0370690 1.00000e-06 0.00643197 0.994952
## 42: 3553.4572: 0.0370676 1.00000e-06 0.00643196 0.994952
##
## Final Estimate of the Negative LLH:
## LLH: -9831.504 norm LLH: -3.553128
## mu omega alpha1 beta1
## 2.938719e-04 6.285312e-11 6.431960e-03 9.949521e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 beta1
## mu -8.092996e+07 1.343625e+11 9.167498e+05 1.977179e+06
## omega 1.343625e+11 -3.361234e+16 -3.473200e+12 -2.218284e+12
## alpha1 9.167498e+05 -3.473200e+12 -3.533850e+07 -4.289392e+07
## beta1 1.977179e+06 -2.218284e+12 -4.289392e+07 -7.180548e+07
## attr(,"time")
## Time difference of 0.03700209 secs
##
## --- END OF TRACE ---
## Warning in sqrt(diag(fit$cvar)): NaNs produced
##
## Time to Estimate Parameters:
## Time difference of 0.2490139 secs
garch_c = garch_f(data_c$returns,data_ch$returns,conf)
##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: norm
## h.start: 2
## llh.start: 1
## Length of Series: 2781
## Recursion Init: mci
## Series Scale: 0.007515565
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -0.58499393 0.5849939 0.05849939 TRUE
## omega 0.00000100 100.0000000 0.10000000 TRUE
## alpha1 0.00000001 1.0000000 0.10000000 TRUE
## gamma1 -0.99999999 1.0000000 0.10000000 FALSE
## beta1 0.00000001 1.0000000 0.80000000 TRUE
## delta 0.00000000 2.0000000 2.00000000 FALSE
## skew 0.10000000 10.0000000 1.00000000 FALSE
## shape 1.00000000 10.0000000 4.00000000 FALSE
## Index List of Parameters to be Optimized:
## mu omega alpha1 beta1
## 1 2 3 5
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 3675.5865: 0.0584994 0.100000 0.100000 0.800000
## 1: 3668.3849: 0.0584981 0.101288 0.113754 0.810783
## 2: 3659.5290: 0.0584967 0.0845767 0.118009 0.807664
## 3: 3600.9398: 0.0584844 0.0361826 0.174126 0.846706
## 4: 3582.7562: 0.0584806 0.0210711 0.188424 0.858273
## 5: 3545.9600: 0.0584691 1.00000e-06 0.178801 0.892129
## 6: 3543.6229: 0.0584666 1.00000e-06 0.173478 0.902708
## 7: 3534.9772: 0.0584640 1.00000e-06 0.161763 0.900971
## 8: 3482.0944: 0.0584437 1.00000e-06 0.0894399 0.938879
## 9: 3462.9763: 0.0584383 1.00000e-06 0.0684187 0.951615
## 10: 3443.9948: 0.0584276 1.00000e-06 0.0267274 0.977655
## 11: 3440.6693: 0.0584276 1.00000e-06 0.0275239 0.978485
## 12: 3440.1856: 0.0584274 1.00000e-06 0.0282446 0.977588
## 13: 3439.2256: 0.0584279 1.00000e-06 0.0303964 0.976776
## 14: 3438.7013: 0.0584283 1.00000e-06 0.0318794 0.975018
## 15: 3438.4746: 0.0584274 1.00000e-06 0.0340236 0.974185
## 16: 3438.4293: 0.0584199 1.00000e-06 0.0343723 0.973537
## 17: 3438.3831: 0.0583993 1.00000e-06 0.0338891 0.973985
## 18: 3438.3800: 0.0583566 1.00000e-06 0.0337647 0.974084
## 19: 3438.3796: 0.0583129 1.00000e-06 0.0337907 0.974093
## 20: 3438.3777: 0.0582692 1.00000e-06 0.0337435 0.974104
## 21: 3438.3717: 0.0578906 1.00000e-06 0.0334763 0.974299
## 22: 3438.3661: 0.0575114 1.00000e-06 0.0334895 0.974225
## 23: 3438.3550: 0.0571323 1.00000e-06 0.0335416 0.974252
## 24: 3438.3328: 0.0556157 1.00000e-06 0.0336053 0.974179
## 25: 3438.3223: 0.0540991 1.00000e-06 0.0336895 0.974151
## 26: 3438.3217: 0.0536471 1.00000e-06 0.0337230 0.974119
## 27: 3438.3217: 0.0536596 1.00000e-06 0.0337176 0.974125
## 28: 3438.3217: 0.0536630 1.00000e-06 0.0337186 0.974125
##
## Final Estimate of the Negative LLH:
## LLH: -10162.93 norm LLH: -3.654417
## mu omega alpha1 beta1
## 4.033077e-04 5.648372e-11 3.371859e-02 9.741245e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 beta1
## mu -93462003.7 -2.458207e+09 -1.928012e+05 -2.336219e+05
## omega -2458206558.4 -2.422838e+15 -4.340546e+10 -6.467782e+10
## alpha1 -192801.2 -4.340546e+10 -1.205340e+06 -1.660497e+06
## beta1 -233621.9 -6.467782e+10 -1.660497e+06 -2.499098e+06
## attr(,"time")
## Time difference of 0.03500199 secs
##
## --- END OF TRACE ---
##
##
## Time to Estimate Parameters:
## Time difference of 0.368021 secs
Plotting of Daily Data
2000-2010 Data Period Variance
# graph variance, VaR for main data
time = c(1:nrow(garch_m))
year = substr(garch_m[1,"DATE"],7,10)
year = as.numeric(year)
time = time/252 + year
data = data.frame(garch_m,time)
data$rm_variance = oneday_m$variance
data$rm_VaR = oneday_m$VaR
plot_ly(data, x = ~time, y = ~variance, name = "Garch Model",
type = "scatter", mode = "lines",
line = list(color = 'rgb(205, 12, 24)', width = 1.5)) %>%
add_trace(y = ~rm_variance, name = "RiskMetrics Model",
line = list(color = 'rgb(22, 96, 167)', width = 1.5)) %>%
layout(title = "Main Data Daily Variance",
xaxis = list(title = "Year"),
yaxis = list(title = "Variance"))
2000-2010 Data Period VaR
plot_ly(data, x = ~time, y = ~VaR, name = "VaR from Garch",
type = "scatter", mode = "lines",
line = list(color = 'rgb(205, 12, 24)', width = 1.5)) %>%
add_trace(y = ~rm_VaR, name = "VaR from RiskMetrics",
line = list(color = 'rgb(22, 96, 167)', width = 1.5)) %>%
layout(title = "Main Data Daily VaR",
xaxis = list(title = "Year"),
yaxis = list(title = "VaR"))
1980-1990 Data Period Variance
# graph variance, VaR for comparison data
time = c(1:nrow(garch_c))
year = substr(garch_c[1,"DATE"],7,10)
year = as.numeric(year)
time = time/252 + year
data = data.frame(garch_c,time)
data$rm_variance = oneday_c$variance
data$rm_VaR = oneday_c$VaR
plot_ly(data, x = ~time, y = ~variance, name = "Garch Model",
type = "scatter", mode = "lines",
line = list(color = 'rgb(205, 12, 24)', width = 1.5)) %>%
add_trace(y = ~rm_variance, name = "RiskMetrics Model",
line = list(color = 'rgb(22, 96, 167)', width = 1.5)) %>%
layout(title = "Comparison Data Daily Variance",
xaxis = list(title = "Year"),
yaxis = list(title = "Variance"))
1980-1990 Data Period VaR
plot_ly(data, x = ~time, y = ~VaR, name = "VaR from Garch",
type = "scatter", mode = "lines",
line = list(color = 'rgb(205, 12, 24)', width = 1.5)) %>%
add_trace(y = ~rm_VaR, name = "VaR from RiskMetrics",
line = list(color = 'rgb(22, 96, 167)', width = 1.5)) %>%
layout(title = "Comparison Data Daily VaR",
xaxis = list(title = "Year"),
yaxis = list(title = "VaR"))